@vstinner
Discussion needed:

During I work on this PR, I've noticed that LMTP does not support the timeout parameter.
See: https://docs.python.org/3.9/library/smtplib.html#smtplib.LMTP

However, LMTP also able to use the socket which is created from SMTP.

IMHO LMTP needs to support the timeout parameter.

>>> import socket
>>> sock = socket.socket(socket.AF_UNIX, socket.SOCK_STREAM)
>>> sock.settimeout(30)
>>> sock.gettimeout()
30.0
>>> b = socket.socket(socket.AF_UNIX, socket.SOCK_STREAM)
>>> b.gettimeout()
>>>

Also AFAIK, Unix Socket also able to set a timeout which is used for LMTP protocol
If you agree, I would like to create an issue for this.

IMHO LMTP needs to support the timeout parameter.

If you think so, please open a separated issue. It's not directly related to https://bugs.python.org/issue39259

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